390 CHAPTER 14. ANALYSIS OF LINEAR TRANSFORMATIONS
Proposition 14.1.3 Suppose A is invertible, b ̸= 0, Ax = b, and (A+B)x1 = b1 where||B||< 1/
∣∣∣∣A−1∣∣∣∣. Then
||x1− x||||x||
≤∥∥A−1
∥∥∥A∥1−∥A−1B∥
(∥b1−b∥∥b∥
+∥B∥∥A∥
)Proof: This follows from the above lemma.
∥x1− x∥∥x∥
=
∥∥∥(I +A−1B)−1 A−1b1−A−1b
∥∥∥∥A−1b∥
≤ 11−∥A−1B∥
∥∥A−1b1−(I +A−1B
)A−1b
∥∥∥A−1b∥
≤ 11−∥A−1B∥
∥∥A−1 (b1−b)∥∥+∥∥A−1BA−1b
∥∥∥A−1b∥
≤∥∥A−1
∥∥1−∥A−1B∥
(∥b1−b∥∥A−1b∥
+∥B∥)
because A−1b/∥∥A−1b
∥∥ is a unit vector. Now multiply and divide by ∥A∥ . Then
≤∥∥A−1
∥∥∥A∥1−∥A−1B∥
(∥b1−b∥∥A∥∥A−1b∥
+∥B∥∥A∥
)≤
∥∥A−1∥∥∥A∥
1−∥A−1B∥
(∥b1−b∥∥b∥
+∥B∥∥A∥
). ■
This shows that the number,∣∣∣∣A−1
∣∣∣∣ ||A|| , controls how sensitive the relative change inthe solution of Ax = b is to small changes in A and b. This number is called the conditionnumber. It is bad when this number is large because a small relative change in b, forexample could yield a large relative change in x.
Recall that for A an n× n matrix, ||A||2 = σ1 where σ1 is the largest singular value.The largest singular value of A−1 is therefore, 1/σn where σn is the smallest singular valueof A. Therefore, the condition number is controlled by σ1/σn, the ratio of the largest to thesmallest singular value of A provided the norm is the usual Euclidean norm.
14.2 The Spectral RadiusEven though it is in general impractical to compute the Jordan form, its existence is all thatis needed in order to prove an important theorem about something which is relatively easyto compute. This is the spectral radius of a matrix.
Definition 14.2.1 Define σ (A) to be the eigenvalues of A. Also,
ρ (A)≡max(|λ | : λ ∈ σ (A))
The number, ρ (A) is known as the spectral radius of A.